The Basics of Two-Phase Flow

Assignment :

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Two Stages of Science

1. Description and Classification (Organization)

• Reflected in basic Flow regime maps for two-phase flow
• Also seen in two-phase heat transfer with the boiling curve

2. Predictive Theories

• Kaye Lathrop once identified three phases in the history of theoretical science:

There are three historical phases in the development of theoretical science. The first beginning at the dawn of history, and only fading in recent years is "back-of-the-envelop science". All predictions were made with hand calculations. The next phase, "science by computer," began in the 1950's with the advent of useful programmable computers. The third and terminal phase of science became noticeable a couple decades ago. It is "science by view graph," and has probably been driven by the dominance of government funding of scientific research.

• Theories that produce useful results with hand calculations are only useful for very simple problems
• Most systems of interest to us are so complex that they need computers to carry out the calculations
• Computers don't recognize differential equations
• A layer of numerical models for the differential equations must be provided

Three Tasks of Numerical Simulation

• Understand the Predictive Theories of Real World relevant to what you need to simulate
  
• Create a Virtual World on the computer that mimics the desired portion of the real world
• Quantify Differences Between the Real and Virtual Worlds

Nomenclature

• Permits clearer and more concise discussions
• Great for job security
• Key terms (see the list of acronyms and  nomenclature)
• Subscripts l or script l (liquid), g (gas), v (vapor), m (mixture), s (saturation), w (wall), I (interface)
• Void fraction ( Greek letter alpha)
• mean density
• mean velocity
  
• relative velocity
• phasic volumetric flows or fluxes
  
• phasic mass flows or fluxes
• saturation temperature and saturation pressure
  
• Quality (x)
• mass quality = vapor mass/ total mass
• flow quality = vapor flow / total flow
• equilibrium quality x = (h_tot-h_ls)/(h_gs-h_ls) here h is the specific enthalpy

Simple Two-Phase Flow Regimes

• Consider horizontal and vertical pipes, most research characterizes either horizontal or vertical flow, although some is available for inclined pipes.

• Flow regimes are frequently plotted on  void fraction vs. mass flux (kg/m**2/s) axes. 
  

• Above 2700 kg/m²s the regime is also called "churn turbulent"

• Slug flow can exist roughly in the void fraction range from 0.3 through 0.5 with mass flow < 2000.. This is basically flow where the bubble diameter reaches the pipe diameter.  Important scaling issues occur here.  Smaller scale experiments can go into slug flow regimes that would not be seen in a full reactor coolant pipe.
  

• There is an additonal flow regime for  horizontal pipes.
  
• What is the flow like in a culvert under a road or a sewer pipe?
• This is horizontally stratified flow, and the conditions for transfer between this regime and standard bubbly or droplet flow are in part related to the Froude number (F=sqrt(V**2/(gD)). This is basically the square root of the ratio of specific kinetic energy to specific gravitational potential energy.
• When we add the complication of heated walls later, more will have to be said about flow regimes.

Physical Feel for void fractions (see my sample void calculations)

• What do you think is the void fraction in a heavy rain storm? Can you think of a way to estimate the void fraction? Think about the distance at which you can no longer see anything. As you look through a volume of air that has a cross sectional area A (area of your iris opening) and viewing length l, the area blocked by rain drops is roughly equal to the product of the number density of drops, the viewing volume, and the projected area of one average drop (pi times the drop radius squared). The maximum range of vision is approximately at the point where this blocked area equals A.

• To get what we want, first we calculate number density as a function of alpha and r. Substitute this in the above equation. and solve for void as a function of l and r.

• In a heavy rain I can't see much past my front fence line which is about 1000 feet away. For lack of a better guess lets take 1/4" drops. We get a void fraction of 0.999972.

• Rearrange the final equation to get visibility as a function of void and r. What does this tell you about fog?

• Calculate void fraction with bubbles just touching. It is about 0.524

• Calculate void fraction where droplets have a surface separation of one diameter. If you assume the drops are equal size and centered in a uniform cubic lattice, the answer is 0.935.

• Uniform droplets will have a surface separation of 1.279 diameters when the void fraction is 0.75.

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